Bernstein-Durrmeyer operators with arbitrary weight functions

Transcription

1 Bernstein-Durrmeyer operators with arbitrary weight functions Elena E. Berdysheva German University of Technology in Oman Muscat, Sultanate of Oman Bernstein-Durrmeyer operators with arbitrary weight functions p. 1/32

2 Bernstein basis polynomials Standard simplex in R d : S d := {x = (x 1,...,x d ) R d : 0 x 1,...,x d 1, x 1 + +x d 1}. Bernstein-Durrmeyer operators with arbitrary weight functions p. 2/32

3 Bernstein basis polynomials Standard simplex in R d : S d := {x = (x 1,...,x d ) R d : 0 x 1,...,x d 1, x 1 + +x d 1}. Barycentric coordinates: x = (x 0,x 1,...,x d ), x 0 := 1 x 1 x d. Bernstein-Durrmeyer operators with arbitrary weight functions p. 2/32

4 Bernstein basis polynomials Standard simplex in R d : S d := {x = (x 1,...,x d ) R d : Barycentric coordinates: 0 x 1,...,x d 1, x 1 + +x d 1}. x = (x 0,x 1,...,x d ), x 0 := 1 x 1 x d. The d-variate Bernstein basis polynomials of degree n are defined by ( ) n B α (x) := x α n! = α α 0!α 1!...α d! xα 0 0 xα 1 1 xα d d, α = (α 0,α 1,...,α d ) N d+1 0 with α := α 0 +α 1 + +α d = n. Bernstein-Durrmeyer operators with arbitrary weight functions p. 2/32

5 Bernstein basis polynomials One-dimensional case: n N and k = 0,1,...,n, ( ) n p n,k (x) := x k (1 x) n k, x [0,1]. k Bernstein-Durrmeyer operators with arbitrary weight functions p. 3/32

6 Bernstein basis polynomials One-dimensional case: n N and k = 0,1,...,n, ( ) n p n,k (x) := x k (1 x) n k, x [0,1]. k 1 p 3,0 = (1 x) x Bernstein-Durrmeyer operators with arbitrary weight functions p. 3/32

7 Bernstein basis polynomials One-dimensional case: n N and k = 0,1,...,n, ( ) n p n,k (x) := x k (1 x) n k, x [0,1]. k p 3,0 = (1 x) 3 p 3,1 = 3x(1 x) x Bernstein-Durrmeyer operators with arbitrary weight functions p. 4/32

8 Bernstein basis polynomials One-dimensional case: n N and k = 0,1,...,n, ( ) n p n,k (x) := x k (1 x) n k, x [0,1]. k p 3,0 = (1 x) 3 p 3,1 = 3x(1 x) 2 p 3,2 = 3x 2 (1 x) x Bernstein-Durrmeyer operators with arbitrary weight functions p. 5/32

9 Bernstein basis polynomials One-dimensional case: n N and k = 0,1,...,n, ( ) n p n,k (x) := x k (1 x) n k, x [0,1]. k p 3,0 = (1 x) 3 p 3,1 = 3x(1 x) 2 p 3,2 = 3x 2 (1 x) p 3,3 = x x Bernstein-Durrmeyer operators with arbitrary weight functions p. 6/32

10 Bernstein basis polynomials Clearly, α =n B α (x) = 1. Bernstein-Durrmeyer operators with arbitrary weight functions p. 7/32

11 Bernstein basis polynomials Clearly, α =n B α (x) = 1. The polynomials {B α } α =n constitute a basis of the space of d-variate algebraic polynomials of total degree n. Bernstein-Durrmeyer operators with arbitrary weight functions p. 7/32

12 Clearly, Bernstein basis polynomials α =n B α (x) = 1. The polynomials {B α } α =n constitute a basis of the space of d-variate algebraic polynomials of total degree n. The Bernstein operator is defined for f C(S d ) by ( Bn f ) ( α (x) := f B α (x). n) α =n Bernstein-Durrmeyer operators with arbitrary weight functions p. 7/32

13 Clearly, Bernstein basis polynomials α =n B α (x) = 1. The polynomials {B α } α =n constitute a basis of the space of d-variate algebraic polynomials of total degree n. The Bernstein operator is defined for f C(S d ) by ( Bn f ) ( α (x) := f B α (x). n) α =n This is a positive linear operator that reproduces linear functions. Uniform convergence for every function in C(S d ). Bernstein-Durrmeyer operators with arbitrary weight functions p. 7/32

14 Bernstein-Durrmeyer operator A similar construction for integrable functions? Bernstein-Durrmeyer operators with arbitrary weight functions p. 8/32

15 Bernstein-Durrmeyer operator A similar construction for integrable functions? Definition. The Bernstein-Durrmeyer operator is defined for f L q (S d ), 1 q <, or f C(S d ) by ( Mn f ) f(y) B S (x) := d α (y) dy B α (x). B S d α (y) dy α =n Bernstein-Durrmeyer operators with arbitrary weight functions p. 8/32

16 Bernstein-Durrmeyer operator A similar construction for integrable functions? Definition. The Bernstein-Durrmeyer operator is defined for f L q (S d ), 1 q <, or f C(S d ) by ( Mn f ) f(y) B S (x) := d α (y) dy B α (x). B S d α (y) dy α =n M n is a positive linear operator that reproduces constant functions. Convergence in L q (S d ), 1 q <, and in C(S d ). Bernstein-Durrmeyer operators with arbitrary weight functions p. 8/32

17 Bernstein-Durrmeyer operator A similar construction for integrable functions? Definition. The Bernstein-Durrmeyer operator is defined for f L q (S d ), 1 q <, or f C(S d ) by ( Mn f ) f(y) B S (x) := d α (y) dy B α (x). B S d α (y) dy α =n M n is a positive linear operator that reproduces constant functions. Convergence in L q (S d ), 1 q <, and in C(S d ). Introduced in the one-dimensional case by Durrmeyer (1967) and, independently, by Lupaş (1972). Bernstein-Durrmeyer operators with arbitrary weight functions p. 8/32

18 Bernstein-Durrmeyer operator A similar construction for integrable functions? Definition. The Bernstein-Durrmeyer operator is defined for f L q (S d ), 1 q <, or f C(S d ) by ( Mn f ) f(y) B S (x) := d α (y) dy B α (x). B S d α (y) dy α =n M n is a positive linear operator that reproduces constant functions. Convergence in L q (S d ), 1 q <, and in C(S d ). Introduced in the one-dimensional case by Durrmeyer (1967) and, independently, by Lupaş (1972). Became known due to Derriennic (starting from 1981). Bernstein-Durrmeyer operators with arbitrary weight functions p. 8/32

19 Bernstein-Durrmeyer operator A similar construction for integrable functions? Definition. The Bernstein-Durrmeyer operator is defined for f L q (S d ), 1 q <, or f C(S d ) by ( Mn f ) f(y) B S (x) := d α (y) dy B α (x). B S d α (y) dy α =n M n is a positive linear operator that reproduces constant functions. Convergence in L q (S d ), 1 q <, and in C(S d ). Introduced in the one-dimensional case by Durrmeyer (1967) and, independently, by Lupaş (1972). Became known due to Derriennic (starting from 1981). Extension to functions on the d-dimensional simplex: Derriennic (starting from 1985). Bernstein-Durrmeyer operators with arbitrary weight functions p. 8/32

20 Weighted Bernstein-Durrmeyer operator Let ρ be a non-negative bounded (regular) Borel measure on S d such that supp(ρ)\( S d ). Bernstein-Durrmeyer operators with arbitrary weight functions p. 9/32

21 Weighted Bernstein-Durrmeyer operator Let ρ be a non-negative bounded (regular) Borel measure on S d such that supp(ρ)\( S d ). L q ρ(s d ), 1 q < : the weighted L q -space with the norm ( ) 1/q f L q ρ := f(x) q dρ(x). S d Bernstein-Durrmeyer operators with arbitrary weight functions p. 9/32

22 Weighted Bernstein-Durrmeyer operator Let ρ be a non-negative bounded (regular) Borel measure on S d such that supp(ρ)\( S d ). L q ρ(s d ), 1 q < : the weighted L q -space with the norm ( ) 1/q f L q ρ := f(x) q dρ(x). S d Definition. The Bernstein-Durrmeyer operator with respect to the measure ρ is defined for f L q ρ(s d ), 1 q <, or f C(S d ) by (M n,ρ f)(x) := α =n f(y) B S d α (y) dρ(y) B S d α (y) dρ(y) B α (x). Bernstein-Durrmeyer operators with arbitrary weight functions p. 9/32

23 Weighted Bernstein-Durrmeyer operator Let ρ be a non-negative bounded (regular) Borel measure on S d such that supp(ρ)\( S d ). L q ρ(s d ), 1 q < : the weighted L q -space with the norm ( ) 1/q f L q ρ := f(x) q dρ(x). S d Definition. The Bernstein-Durrmeyer operator with respect to the measure ρ is defined for f L q ρ(s d ), 1 q <, or f C(S d ) by (M n,ρ f)(x) := α =n f(y) B S d α (y) dρ(y) B S d α (y) dρ(y) B α (x). M n,ρ is a positive linear operator that reproduces constant functions. Bernstein-Durrmeyer operators with arbitrary weight functions p. 9/32

24 Jacobi weights The weighted Bernstein-Durrmeyer operator M n,ρ is very well studied for Jacobi weights, i.e., for dρ(x) = x µ dx, with µ = (µ 0,µ 1,...,µ d ) R d+1, where µ i > 1, i = 0,1,...,d. Bernstein-Durrmeyer operators with arbitrary weight functions p. 10/32

25 Jacobi weights The weighted Bernstein-Durrmeyer operator M n,ρ is very well studied for Jacobi weights, i.e., for dρ(x) = x µ dx, with µ = (µ 0,µ 1,...,µ d ) R d+1, where µ i > 1, i = 0,1,...,d. Bernstein-Durrmeyer operators with respect to Jacobi weights were introduced by Păltănea (1983), Berens and Xu (1991), in the multivariate case by Ditzian (1995). Bernstein-Durrmeyer operators with arbitrary weight functions p. 10/32

26 Jacobi weights The weighted Bernstein-Durrmeyer operator M n,ρ is very well studied for Jacobi weights, i.e., for dρ(x) = x µ dx, with µ = (µ 0,µ 1,...,µ d ) R d+1, where µ i > 1, i = 0,1,...,d. Bernstein-Durrmeyer operators with respect to Jacobi weights were introduced by Păltănea (1983), Berens and Xu (1991), in the multivariate case by Ditzian (1995). They were studied by many authors, e.g., Derriennic, Berens, Xu, Ditzian, Chen, Ivanov, X.-L. Zhou, Knoop, Gonska, Heilmann, Abel, Jetter, Stöckler,... Bernstein-Durrmeyer operators with arbitrary weight functions p. 10/32

27 Jacobi weights Berens and Xu noticed that the Bernstein-Durrmeyer operator with respect to Jacobi weight is a summation method for Jacobi series. Bernstein-Durrmeyer operators with arbitrary weight functions p. 11/32

28 Jacobi weights Berens and Xu noticed that the Bernstein-Durrmeyer operator with respect to Jacobi weight is a summation method for Jacobi series. This follows from the spectral properties of the Bernstein-Durrmeyer operators with respect to Jacobi weights: Bernstein-Durrmeyer operators with arbitrary weight functions p. 11/32

29 Jacobi weights Berens and Xu noticed that the Bernstein-Durrmeyer operator with respect to Jacobi weight is a summation method for Jacobi series. This follows from the spectral properties of the Bernstein-Durrmeyer operators with respect to Jacobi weights: the eigenfunctions in this case are Jacobi polynomials. Bernstein-Durrmeyer operators with arbitrary weight functions p. 11/32

30 Jacobi weights Berens and Xu noticed that the Bernstein-Durrmeyer operator with respect to Jacobi weight is a summation method for Jacobi series. This follows from the spectral properties of the Bernstein-Durrmeyer operators with respect to Jacobi weights: the eigenfunctions in this case are Jacobi polynomials. For d = 1, µ 1 = µ 0 = 1 2, it is exactly the de la Vallée-Poussin mean for Chebyshev series (1908). Bernstein-Durrmeyer operators with arbitrary weight functions p. 11/32

31 Jacobi weights Berens and Xu noticed that the Bernstein-Durrmeyer operator with respect to Jacobi weight is a summation method for Jacobi series. This follows from the spectral properties of the Bernstein-Durrmeyer operators with respect to Jacobi weights: the eigenfunctions in this case are Jacobi polynomials. For d = 1, µ 1 = µ 0 = 1 2, it is exactly the de la Vallée-Poussin mean for Chebyshev series (1908). The univariate ultraspherical case (µ 1 = µ 0 ) was studied already by Kogbetliantz (1922), Bernstein-Durrmeyer operators with arbitrary weight functions p. 11/32

32 Jacobi weights Berens and Xu noticed that the Bernstein-Durrmeyer operator with respect to Jacobi weight is a summation method for Jacobi series. This follows from the spectral properties of the Bernstein-Durrmeyer operators with respect to Jacobi weights: the eigenfunctions in this case are Jacobi polynomials. For d = 1, µ 1 = µ 0 = 1 2, it is exactly the de la Vallée-Poussin mean for Chebyshev series (1908). The univariate ultraspherical case (µ 1 = µ 0 ) was studied already by Kogbetliantz (1922), the general univariate case by Bavinck (1976). Bernstein-Durrmeyer operators with arbitrary weight functions p. 11/32

33 Motivation: Learning Theory In a joint paper with Kurt Jetter (JAT, 2010), we started to study the multivariate Bernstein-Durrmeyer operators with respect to general measure. Bernstein-Durrmeyer operators with arbitrary weight functions p. 12/32

34 Motivation: Learning Theory In a joint paper with Kurt Jetter (JAT, 2010), we started to study the multivariate Bernstein-Durrmeyer operators with respect to general measure. We were motivated by paper D.-X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math. 25 (2006), Bernstein-Durrmeyer operators with arbitrary weight functions p. 12/32

35 Motivation: Learning Theory In a joint paper with Kurt Jetter (JAT, 2010), we started to study the multivariate Bernstein-Durrmeyer operators with respect to general measure. We were motivated by paper D.-X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math. 25 (2006), They considered the univariate M n,ρ and used it for estimates for SVM classifiers. Bernstein-Durrmeyer operators with arbitrary weight functions p. 12/32

36 Motivation: Learning Theory In a joint paper with Kurt Jetter (JAT, 2010), we started to study the multivariate Bernstein-Durrmeyer operators with respect to general measure. We were motivated by paper D.-X. Zhou and K. Jetter, Approximation with polynomial kernels and SVM classifiers, Adv. Comput. Math. 25 (2006), They considered the univariate M n,ρ and used it for estimates for SVM classifiers. Recently (2012), Bing-Zheng Li used the multivariate operators M n,ρ to obtain estimates for learning rates of least-square regularized regression with polynomial kernels. Bernstein-Durrmeyer operators with arbitrary weight functions p. 12/32

37 Motivation: Learning Theory Typical problems in learning theory: Regression. E.g., least squares x Bernstein-Durrmeyer operators with arbitrary weight functions p. 13/32

38 Motivation: Learning Theory Typical problems in learning theory: Regression. E.g., least squares x Classification. Bernstein-Durrmeyer operators with arbitrary weight functions p. 13/32

39 Convergence: examples Numerical experiments show that convergence holds for a wide class of measures ρ. Bernstein-Durrmeyer operators with arbitrary weight functions p. 14/32

40 Convergence: examples Numerical experiments show that convergence holds for a wide class of measures ρ. Example 1. Consider the measure dρ(x) = w(x) dx with ( w(x) = x 1 ) y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 14/32

41 Example 1 dρ(x) = ( x 1 2) dx, f(x) = x, n = 5 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 15/32

42 Example 1 dρ(x) = ( x 1 2) dx, f(x) = x, n = 20 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 16/32

43 Example 1 dρ(x) = ( x 1 2) dx, f(x) = x, n = 100 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 17/32

44 Example 1 dρ(x) = ( x 1 2) dx, f(x) = x, n = 500 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 18/32

45 Convergence: examples On the other hand, it is not difficult to construct an example of an operator for which convergence in C([0, 1]) fails. Bernstein-Durrmeyer operators with arbitrary weight functions p. 19/32

46 Convergence: examples On the other hand, it is not difficult to construct an example of an operator for which convergence in C([0, 1]) fails. Example 2. dρ(x) = w(x)dx with { 1, 0 x 1 w(x) = 2, 1 0, 2 < x x Bernstein-Durrmeyer operators with arbitrary weight functions p. 19/32

47 Convergence: examples On the other hand, it is not difficult to construct an example of an operator for which convergence in C([0, 1]) fails. Example 2. dρ(x) = w(x)dx with { 1, 0 x 1 w(x) = 2, 1 0, 2 < x The Bernstein-Durrmeyer operator has the form (M n,ρ f)(x) = n k= f(y)yk (1 y) n k dy yk (1 y) n k dy x ( ) n x k (1 x) n k. k Bernstein-Durrmeyer operators with arbitrary weight functions p. 19/32

48 Convergence: examples On the other hand, it is not difficult to construct an example of an operator for which convergence in C([0, 1]) fails. Example 2. dρ(x) = w(x)dx with { 1, 0 x 1 w(x) = 2, 1 0, 2 < x The Bernstein-Durrmeyer operator has the form (M n,ρ f)(x) = n k= f(y)yk (1 y) n k dy yk (1 y) n k dy x ( ) n x k (1 x) n k. k Then for f(x) = x we have (M n,ρ f)(x) 1 2, x [0,1]. Bernstein-Durrmeyer operators with arbitrary weight functions p. 19/32

49 Example 2 dρ = χ [ 0, 1 2] dx, f(x) = x, n = 5 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 20/32

50 Example 2 dρ = χ [ 0, 1 2] dx, f(x) = x, n = 15 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 21/32

51 Example 2 dρ = χ [ 0, 1 2] dx, f(x) = x, n = 30 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 22/32

52 Example 2 dρ = χ [ 0, 1 2] dx, f(x) = x, n = 100 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 23/32

53 Example 2 dρ = χ [ 0, 1 2] dx, f(x) = x, n = 200 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 24/32

54 Example 2 dρ = χ [ 0, 1 2] dx, f(x) = x, n = 200 : y x Bernstein-Durrmeyer operators with arbitrary weight functions p. 25/32

55 Convergence in C(S d ) We give necessary and sufficient conditions on the measure ρ for convergence in C(S d ). Bernstein-Durrmeyer operators with arbitrary weight functions p. 26/32

56 Convergence in C(S d ) We give necessary and sufficient conditions on the measure ρ for convergence in C(S d ). Recall that a measure ρ on S d is called strictly positive, if ρ(a S d ) > 0 for every open set A R d with A S d. Bernstein-Durrmeyer operators with arbitrary weight functions p. 26/32

57 Convergence in C(S d ) We give necessary and sufficient conditions on the measure ρ for convergence in C(S d ). Recall that a measure ρ on S d is called strictly positive, if ρ(a S d ) > 0 for every open set A R d with A S d. This is equivalent to the fact that supp(ρ) = S d. Bernstein-Durrmeyer operators with arbitrary weight functions p. 26/32

58 Convergence in C(S d ) We give necessary and sufficient conditions on the measure ρ for convergence in C(S d ). Recall that a measure ρ on S d is called strictly positive, if ρ(a S d ) > 0 for every open set A R d with A S d. This is equivalent to the fact that supp(ρ) = S d. Theorem. (EB, JMAA, 2012) We have lim f M n,ρf C = 0 for all f C(S d ) n if and only if ρ is strictly positive on S d. Bernstein-Durrmeyer operators with arbitrary weight functions p. 26/32

59 Convergence in C(S d ) We say that ρ is a Jacobi-like measure, if dρ(x) = w(x)dx, Bernstein-Durrmeyer operators with arbitrary weight functions p. 27/32

60 Convergence in C(S d ) We say that ρ is a Jacobi-like measure, if dρ(x) = w(x)dx, and there are two exponents ν µ > e and two constants 0 < a,a < such that a x ν w(x) A x µ, x S d. Bernstein-Durrmeyer operators with arbitrary weight functions p. 27/32

61 Convergence in C(S d ) We say that ρ is a Jacobi-like measure, if dρ(x) = w(x)dx, and there are two exponents ν µ > e and two constants 0 < a,a < such that a x ν w(x) A x µ, x S d. Denote ϕ ei (x) = x i, ϕ 2ei (x) = x 2 i. Bernstein-Durrmeyer operators with arbitrary weight functions p. 27/32

62 Convergence in C(S d ) Theorem. (Kurt Jetter-EB, JAT, 2010) Let ρ be a Jacobi-like measure with ν µ < 1. Then ϕ ei M n,ρ (ϕ ei ) C Cn 1 ( ν µ ) 2 and ϕ 2ei M n,ρ (ϕ 2ei ) C Cn 1 ( ν µ ) 2. Bernstein-Durrmeyer operators with arbitrary weight functions p. 28/32

63 Convergence in C(S d ) Theorem. (Kurt Jetter-EB, JAT, 2010) Let ρ be a Jacobi-like measure with ν µ < 1. Then ϕ ei M n,ρ (ϕ ei ) C Cn 1 ( ν µ ) 2 and ϕ 2ei M n,ρ (ϕ 2ei ) C Cn 1 ( ν µ ) 2. Corollary. Let ρ be a Jacobi-like measure with ν µ < 1. Let f C(S d ). Then ) f M n,ρ f C Cω (f,n 1 ( ν µ ) 4, where ω(f,δ) = sup{ f(x) f(t) : t x 2 < δ} denote the modulus of continuity of f. Bernstein-Durrmeyer operators with arbitrary weight functions p. 28/32

64 Convergence onsuppρ Theorem. (EB, 2012) Let x (suppρ). Let f be bounded on suppρ and continuous at x. Then lim f(x) M n,ρf(x) = 0. n Bernstein-Durrmeyer operators with arbitrary weight functions p. 29/32

65 Convergence onsuppρ Theorem. (EB, 2012) Let x (suppρ). Let f be bounded on suppρ and continuous at x. Then lim f(x) M n,ρf(x) = 0. n Theorem. (EB, 2012) Let A be a compact set, A (suppρ). Let f be bounded on suppρ and continuous on A. Then lim n f M n,ρf C(A) = 0. Bernstein-Durrmeyer operators with arbitrary weight functions p. 29/32

66 Convergence in L q ρ(s d ) Theorem. (Bing-Zheng Li, 2012) Let ρ be a non-negative bounded (regular) Borel measure on S d such that supp(ρ)\( S d ). Let 1 q <. Then for every f L q ρ(s d ). lim f M n,ρf L q n ρ = 0 Bernstein-Durrmeyer operators with arbitrary weight functions p. 30/32

67 Convergence in L q ρ(s d ) Consider the K-functional K q (f,t) = inf{ f g L q ρ +t max i=1,...,d ig C : g C 1 (S d )}. Bernstein-Durrmeyer operators with arbitrary weight functions p. 31/32

68 Convergence in L q ρ(s d ) Consider the K-functional K q (f,t) = inf{ f g L q ρ +t max i=1,...,d ig C : g C 1 (S d )}. Theorem. (Bing-Zheng Li-EB, 2012) Let ρ be a non-negative bounded Borel measure on S d such that suppρ\ S d, and let f L q ρ(s d ), 1 q <. Then ( f M n,ρ f L q ρ 2K q f, C q d [ ) ρ(s d ) ]1 q, 1 q <, n where C q is a constant that depends only on q. Moreover, C q = 1 for 1 q 2. Bernstein-Durrmeyer operators with arbitrary weight functions p. 31/32

69 Convergence in L q ρ(s d ) Consider the K-functional K q (f,t) = inf{ f g L q ρ +t max i=1,...,d ig C : g C 1 (S d )}. Theorem. (Bing-Zheng Li-EB, 2012) Let ρ be a non-negative bounded Borel measure on S d such that suppρ\ S d, and let f L q ρ(s d ), 1 q <. Then ( f M n,ρ f L q ρ 2K q f, C q d [ ) ρ(s d ) ]1 q, 1 q <, n where C q is a constant that depends only on q. Moreover, C q = 1 for 1 q 2. Remark: the cases q = 1, q = 2 are due to Bing-Zheng Li. Bernstein-Durrmeyer operators with arbitrary weight functions p. 31/32

70 Thank you for your attention! Bernstein-Durrmeyer operators with arbitrary weight functions p. 32/32